Our goal is to predict the band structure of photonic crystals. This task requires us to compute a number of the smallest non-zero eigenvalues of the time-harmonic Maxwell operator depending on the chosen Bloch boundary conditions. We propose to use a block inverse iteration preconditioned with a suitably modified geometric multigrid method. Since we are only interested in non-zero eigenvalues, we eliminate the large null space by combining a lifting operator and a secondary multigrid method. To obtain suitable initial guesses for the iteration, we employ a generalized extrapolation technique based on the minimization of the Rayleigh quotient that significantly reduces the number of iteration steps and allows us to treat families of very large eigenvalue problems efficiently.

翻译：我们的目标是预测光子晶体的能带结构。该任务需要根据所选的布洛赫边界条件，计算时谐麦克斯韦算子中若干最小的非零特征值。我们提出采用经适当修正的几何多重网格方法进行预处理的块逆迭代法。由于仅关注非零特征值，我们通过结合提升算子与辅助多重网格方法消除大零空间。为获得迭代所需的合适初始猜测，我们采用基于瑞利商最小化的广义外推技术，该方法可显著减少迭代步数，并使我们能够高效处理大规模特征值问题族。